Suppose that consumers see all tomatoes as identical and that there are many tom

Question

Suppose that consumers see all tomatoes as identical and that there are many tomato producers in the market. The current market equilibrium price of tomatoes is $2.30 per pound. Pedro's small hydroponic tomato farm has the following short-run total cost curve: TC = 10 + 3.8Q - 0.9Q^2 + 0.1Q^3 Where TC is the total cost in thousands of dollars and Q is the quantity of tomatoes produced in thousands of pounds per month. Pedro's marginal cost curve (which can be derived from the TC curve) is MC = 3.8 - 1.8Q + 0.3Q^2 a. What quantity of tomatoes should Pedro produce in the short run to maximize his profit? b. If the farm maximizes profit, how much profit will it earn per month? c. Use calculus to show that marginal cost is as given. d. Express profit as a function of Q, use calculus to maximize this function, and show that the solution is the same as your answer to (a)

Explanation / Answer

A) In order to maximize profit in perfectly competitive market, set P = MC

That is, 2.3 = 3.8 - 1.8Q + 0.3Q2

Or, 0.3Q2 - 1.8Q + 1.5 = 0

Or, Q* = 5 or 1

B) Profit = PQ - TC

When Q = 5,

Profit = (2.3*5) - (10 + 3.8(5) - 0.9(5*5) + 0.1(5*5*5) = 11.5 - 19 = -$7.5

When Q = 1,

Profit = (2.3*1) - (10 + 3.8(1) - 0.9(1*1) + 0.1(1*1*1)) = 2.3 - 13 = -$10.7

Thus, Q = 5 must be chosen

C)

MC = dTC/dQ = d(10 + 3.8Q - 0.9Q2 + 0.1Q3)/dQ = 3.8 - 0.9(2Q) + 0.1(3Q) = 3.8 - 1.8Q + 0.3Q

D)

Profit = TR - TC = 2.3Q - (10 + 38Q - 0.9Q^2 + 0.1Q^3)

dProfit/dQ = 2.3 - 3.8 + 1.8Q - 0.3Q2= 0

Or, 0.3Q2 - 1.8Q + 1.5 = 0

Solving this gives same result as (a)

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